The present invention relates to an assist system (CAD system) for designing a substance or a material using a computer and, more particularly, to MM (Molecular Mechanics) calculation and MD (Molecular Dynamics) calculation.
As is generally known, when a substance or a material is to be designed using a computer, the structural stability or dynamic behavior of a crystal system is analyzed using MM (Molecular Mechanics) calculation or MD (Molecular Dynamics) calculation.
In MD calculation, a change over time in motion of atoms or molecules at a finite temperature is tracked in accordance with a Newton's equation. This is one of essential techniques for substance/material design as a means for checking the microscopic behavior of a system (reference [1]: Akira Ueda, "Computer Simulation", Asakura Shoten (1990)).
MM calculation is a method of analyzing the most stable structure of a system and corresponds to MD calculation at the absolute zero (T=0 K) (reference [2]: Isao Okada and Eiji Osawa, "Introduction to Molecular Simulation", Kaibundo (1989)).
The MM calculation and MD calculation can be applied to both an infinite system (e.g., a crystal) and an isolated system (e.g., a molecule). The MM/MD calculation of the present invention is directed to the infinite system.
The "infinite system" means a system having a sufficient range at the atomic/molecular level, i.e., a solid or liquid such as a crystal, an amorphous substance, or a liquid crystal. Since such an infinite system cannot be directly handled by computer simulation, a cell (unit cell) having a predetermined size and called a unit cell is set in the infinite system. Accordingly, the infinite system is perceived as a system in which the unit cells are periodically and three-dimensionally arranged.
With this arrangement, computer simulation for the infinite system can be performed only by performing calculation while setting a periodic boundary condition for the unit cell.
This technique using the periodic boundary condition is supposed to be adequate for a crystal system having a periodicity. However, this method is also applied, as an approximation method, to a system whose periodicity disappears because of impurities.
The impurity is the major factor (disorder) for canceling the periodicity. The impurity here means a substitutional/interstitial impurity atom or a defect (the three types of disorders will be referred to as "impurity atoms" hereinafter). A set of one or more impurity atoms existing relatively close to each other in a crystal is called an "impurity". In addition to the impurity, a destruction phenomenon or the like can also be considered as a disorder. In this specification, an infinite system in which such disorders are distributed at a relatively low concentration (a system in which there are almost no interactions between disorders) is defined as an "aperiodic system".
When the disorder is an impurity, the "aperiodic system" can be regarded as a "system containing an isolated impurity". The "system containing an isolated impurity" means an infinite system having only one "impurity", or an infinite system having a plurality of impurities which are sufficiently separated from each other and do not interact each other.
In other words, in such a system, the content of the "impurity" is very low. An example is a defect in a (compound) semiconductor. This defect is known as a DX center which gives a deep impurity level near the center of the energy gap of the semiconductor and largely affects the device characteristics of a semiconductor laser or the like.
In analyzing the structural stability or dynamic behavior of such an "aperiodic system" on the basis of the above-described MM calculation or MD calculation, a method of performing simulation while giving prominence on one unit cell under the "periodic boundary condition" is generally known as a "supercell method".
In the supercell method, only one disorder is arranged in the unit cell, and a sufficiently large unit cell size is set, thereby approximately describing a state wherein disorders do not interact each other. More specifically, in the supercell method, to realize the state of the above-described aperiodic system (when the disorder is an "impurity", "a system containing an isolated impurity"), normally, only one disorder is set in the unit cell. However, this disorder is automatically arranged in peripheral unit cells (to be referred to as "image cells") because of the periodic boundary condition. To perform proper calculation while minimizing interaction with these disorders, the unit cell size must be as large as possible.
However, in the supercell method, the unit cell size must be determined in advance before the start of simulation (the size cannot be changed during simulation), and the size is often determined (to be relatively small) in consideration of the balance between the calculation accuracy and the calculation time.
When the supercell method is used, the aperiodic system can be handled within the same framework as that of the calculation method for a periodic system. For this reason, the supercell method is generally used as a convenient method for MD calculation or MM calculation.
To perform MD calculation or MM calculation using the supercell method, the force acting on each constituent atom of the system must be calculated in each step of repetitive calculation of MM calculation or at each time (one step of repetitive calculation) of MD calculation on the basis of the interatomic interaction. The potential energy of the system is often simultaneously calculated on the basis of the interatomic interaction. When the interatomic interaction is a long-range interaction, calculation of the force or energy slowly converges. Particularly, for a Coulomb interaction described as an r.sup.-1 function, accurate calculation can hardly be executed when the sum is simply calculated for pairs of atoms, as is known. Therefore, a method called an Ewald method is generally used when calculation of the force or energy according to the Coulomb interaction is performed for the above-described infinite system (a system with the periodic boundary condition) (reference [1]).
This Ewald method can be applied to calculate the lattice sum (the sum for equivalent atoms belonging to different unit cells) of not only the Coulomb interaction but also an interaction given by an r.sup.-n function (multipole interaction or van der Waals force). This is an excellent method because, by appropriately speeding up convergence of calculation of the lattice sum in a real space and an imaginary space, the force or energy of the r.sup.-n long-range interaction is efficiently calculated.
However, MM/MD calculation for a periodic system using the conventional method or for an aperiodic system to which the supercell method is applied has the following problems.
Most of the MM/MD calculation execution time is spent for calculation of the force (and the energy). When the interatomic interaction is a long-range interaction, the number of pairs which must be taken into consideration becomes enormous. Particularly, for the Coulomb interaction, the calculation time still poses a serious problem even when the Ewald method is applied.
For MD calculation, when a temperature fluctuation is to be suppressed within the allowance in terms of statistical mechanics, the MD calculation must be performed for a system in which a number N of atoms per unit cell is at least several hundred. For this reason, it is an important challenge to minimize the amount of calculation of the force.
Especially, for the purpose of designing a substance/material, the MD calculation must be repeatedly performed for a number of systems while changing the simulation conditions including the temperature. Therefore, shortening of the calculation time, i.e., efficient calculation of the force is an important key to the practical new materials design.
In addition, since one step of the repetitive calculation in MD calculation corresponds to a time of 1 picosecond or less (about 0.01 PS) in a real system, the repetitive calculation must be performed in several ten thousand to several hundred thousand or more steps to analyze a change over time in system within a physically significant range. For this reason, for example, several ten days are still necessary to perform MD calculation once regardless of the improvement of the computer capability.
The above problems of the calculation time are posed when the long-range interaction is taken into consideration for both the periodic system and the aperiodic system handled by the supercell method. MM/MD calculation for the aperiodic system also has the following problems.
When the aperiodic system is to be handled by the supercell method, calculation must be performed using a unit cell with a sufficiently large size such that interaction between disorders contained in different unit cells can be neglected, as described above.
However, when the interatomic interaction is a long-range interaction, it is practically difficult to execute calculation using a sufficiently large unit cell because of the balance between the calculation time and the accuracy, resulting in a calculation error depending on the cell size. More specifically, in a normal simulation, calculation is often executed using a unit cell with a relatively small size in order to shorten the calculation time. In this case, the interaction with disorders contained in the peripheral unit cells cannot be neglected depending on the unit cell size, and a calculation error corresponding to the unit cell size is generated. In other words, the situation of the "aperiodic system" defined above cannot be exactly handled.
To confirm the dependency on the unit cell size, MM/MD calculation must be performed while changing the unit cell size, and the results must be compared. The Coulomb interaction as the most representative interaction is described by an r.sup.-1 function. For this reason, to confirm convergence of the energy at, e.g., a 10-times higher accuracy, MM/MD calculation must be performed using a unit cell whose number N of atoms is 10.sup.3 times larger.
However, since the calculation time increases in proportion to the square of N, as described above, even confirmation of the dependency on the unit cell size can hardly be performed for the Coulomb interaction.
When the Coulomb interaction (charges) is considered in the aperiodic system, the following problem is generated in the supercell method. When the sum of charges of local disorders is not 0, the sum of charges per unit cell also deviates from 0 due to introduction of such disorders (e.g., an ion, a substitutional/interstitial impurity atom, and a defect). In the supercell method for which the periodic boundary condition is assumed, the sum of charges per unit cell must absolutely be 0 to prevent divergence of the energy, i.e., a restriction is technically required for the convenience of numerical calculation.
As a result of this restriction, when a system containing a disorder which breaks the charge neutrality is to be handled by the supercell method, and the sum of charges per unit cell is not 0, artificial charges (e.g., uniform charges) must be added to each constituent atom to nullify the sum. This disables calculation using the real charge distribution in the system. Particularly, consideration of the artificial charges largely affects the potential energy value.
The problems of the conventional method of analyzing the periodic system or aperiodic system using the supercell method will be summarized below.
(1) In MM/MD calculation, most of the MM/MD calculation execution time is spent for calculation of the force (and the energy). When the interatomic interaction is a long-range interaction, the number of pairs which must be taken into consideration becomes enormous. Particularly, for the Coulomb interaction, the calculation time still poses a serious problem even when the Ewald method is applied.
(2) When calculation for the aperiodic system is to be performed on the basis of the supercell method, calculation must be performed using a unit cell with a sufficiently large size such that interaction between disorders contained in different unit cells can be neglected.
However, when the interatomic interaction is a long-range interaction, it is practically difficult to execute calculation using a sufficiently large unit cell because of the calculation time, resulting in a calculation error depending on the cell size.
To confirm the dependency on the unit cell size, MM/MD calculation must be performed while changing the unit cell size, and the results must be compared. For a pair potential, the calculation time increases in proportion to the square of the number N of atoms. Therefore, for the r.sup.-n long-range interaction, even confirmation of the dependency on the unit cell size can hardly be performed.
(3) When analysis using the supercell method is to be performed for a system where the charge neutrality has been lost, the charge distribution must be corrected to nullify the total charges per unit cell. In this case, the total potential energy, the equilibrium lattice constant, the equilibrium atomic coordinates, and the like of the system undesirable change, so the charge distribution of the system where the charge neutrality has been lost cannot be strictly handled.